Molecular orbital theory
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In chemistry, molecular orbital (MO) theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule. Because electrons are the fundamental constituents of matter involved in bonding, their involvement in bonding has been studied exhaustively by chemists. Electrons are shared among individual atoms in a molecule to form covalent chemical bonds. Generally, up to three bonds can form between atoms in a molecule. Single, or sigma covalent bonds result from the interaction between the nuclei of two discrete atoms; multiple bonds then can result due to the additional formation of pi bonds between overlapping orbitals of like symmetries. Electrons in sigma bonds are located between the nuclei, while electrons in pi bonds are delocalized in regions above and below the nuclei.
The spatial and energetic properties of electrons within atoms are fixed by quantum mechanics to form orbitals that contain these electrons. While atomic orbitals contain electrons ascribed to a single atom, molecular orbitals, which surround a number of atoms in a molecule, contain valence electrons between atoms. Molecular orbital theory, which was proposed in the early twentieth century, revolutionized the study of bonding by approximating the positions of bonded electrons—the molecular orbitals—as Linear Combinations of Atomic Orbitals (LCAO). These approximations are made by applying the Density Functional Theory (DFT) and Hartree–Fock (HF) models to the Schrödinger equation.
MO theory applies the wavelike behavior of electrons as predicted by quantum mechanics, in that electrons no longer deterministically are given defined coordinates, but rather are given probable locations according to the mathematical wavefunctions defining all the possible positions of the electrons. These wave functions, or electron eigenstates, quantitatively describe the atomic orbital basis in which an electron temporarily can reside. Molecular orbitals result from the mixing of these atomic orbitals. In this theory, each molecule has a set of molecular orbitals, in which it is assumed that the molecular orbital wave function ψj can be written as a simple weighted sum of the n constituent atomic orbitals χi, according to the following equation:
Connection to MO diagrams and molecular geometry
Using molecular orbital theory, a molecular orbital diagram can be constructed that features the individual atomic basis orbitals and the molecular orbitals resulting from atomic interactions to form molecules. All atomic s, p, d and f orbitals are assigned relative electronic energies, and orbital interactions are predicted by the relative proximity in the energies of the individual atomic orbitals. The closer the atomic orbitals are in energy, the stronger and more likely their molecular orbital interaction is. Through qualitative molecular orbital analysis through a molecular orbital diagram, the chemist is able to display the resulting molecular orbitals stemming from atomic interaction based on their relative energies and assign bonding, nonbonding and/or antibonding character to each molecular orbital. After these assignments are made, the chemist is able to identify the Highest Occupied Molecular Orbital (HOMO) that contains electrons and the Lowest Unoccupied Molecular Orbital (LUMO) that does not contain electrons. The difference in energy between these two frontier orbitals can be used to predict the strength and stability of transition metal complexes, as well as the colors they produce in solution. The known geometries of all molecules and their (s,p,d,f) orbitals, as predicted by VSEPR theory, are used in conjunction with molecular orbital theory in determining whether select orbitals of an atomic basis are used for bonding, nonbonding, or antibonding interactions within a molecule. Bonding orbitals participate in strengthening the interaction between two atoms of a molecule; nonbonding orbitals exclusively are associated with one atom and usually represent a lone pair; antibonding orbitals participate in weakening the interaction between two atoms of a molecule. Defining this interaction through symmetry adapted linear combinations (SALC's), in which the geometry of valence orbital interaction is illustrated, allows the chemist to justify the assignment of a particular molecular orbital as either bonding, nonbonding or antibonding. SALC's of a molecule can be drawn by knowing either the VSEPR geometry or point group symmetry of that molecule.
One may determine cij coefficients numerically by substituting this equation into the Schrödinger equation and applying the variational principle. The variational principle is a mathematical technique used in quantum mechanics to build up the coefficients of each atomic orbital basis. A larger coefficient means that the orbital basis is composed more of that particular contributing atomic orbital—hence, the molecular orbital is best characterized by that type. This method of quantifying orbital contribution as Linear Combinations of Atomic Orbitals is used in computational chemistry. An additional unitary transformation can be applied on the system to accelerate the convergence in some computational schemes. Molecular orbital theory was seen as a competitor to valence bond theory in the 1930s, before it was realized that the two methods are closely related and that when extended they become equivalent. Valence bond theory further describes the composition of each orbital in a molecule. Unlike MO theory, valence bond theory postulates that molecular orbitals are hybridized and contain a certain percentage of s, p and d character as determined by the number of bonds to the central atom (i.e. tetrahedral complexes feature four sp3 hybridized orbitals that contain 25% s character). It similarly proposes that the strength of bonds increases with increasing overlap between atomic orbitals.
Molecular orbital theory was developed, in the years after valence bond theory had been established (1927), primarily through the efforts of Friedrich Hund, Robert Mulliken, John C. Slater, and John Lennard-Jones. MO theory was originally called the Hund-Mulliken theory. The word orbital was introduced by Mulliken in 1932. By 1933, the molecular orbital theory had been accepted as a valid and useful theory. According to German physicist and physical chemist Erich Hückel, the first quantitative use of molecular orbital theory was the 1929 paper of Lennard-Jones. The first accurate calculation of a molecular orbital wavefunction was that made by Charles Coulson in 1938 on the hydrogen molecule. By 1950, molecular orbitals were completely defined as eigenfunctions (wave functions) of the self-consistent field Hamiltonian and it was at this point that molecular orbital theory became fully rigorous and consistent. This rigorous approach is known as the Hartree–Fock method for molecules although it had its origins in calculations on atoms. In calculations on molecules, the molecular orbitals are expanded in terms of an atomic orbital basis set, leading to the Roothaan equations. This led to the development of many ab initio quantum chemistry methods. In parallel, molecular orbital theory was applied in a more approximate manner using some empirically derived parameters in methods now known as semi-empirical quantum chemistry methods.
Application to ligand field theory
Ligand field theory resulted from combining the principles laid out in molecular orbital theory and crystal field theory, which describes the loss of degeneracy of frontier orbitals in transition metal complexes. Griffith and Orgel championed ligand field theory as a more accurate description of transition metal coordinated compounds. They used the electrostatic principles established in crystal field theory to describe transition metal ions in solutions, and they used molecular orbital theory to explain the differences in transition metal complex interactions. In their paper, they propose that the chief consequence of differences in colors produced by transition metal complexes in solution is incomplete d orbital subshells. That is, the d orbitals of transition metals unoccupied by electrons participate in the bonding that influences the colors they emit in solution. Through ligand field theory, the connection between d orbital energy of transition metal complexes and the character of ligands bound to the transition metal was established. Based on the geometric symmetry between the various d orbitals (dz2, dx2-y2, dxz, dyz, dxy) of the metal and its associated ligands, the relative energies of the d orbitals could be qualitatively and quantitatively assessed. Ligand field theory, therefore, explains how transition metal complexes adjust structurally to reduce degeneracy among d orbital energies. This phenomenon is known as the Jahn-Teller effect. Because d orbital energies are affected differently when surrounded by a field of neighboring ligands, as determined by SALCs, they separately are raised or lowered in energy based on the intensity that their geometries permit them to interact with the ligands. Thus, ligand field theory allows the chemist to determine d orbital energy splittings in transition metal complexes based on the VSEPR-assigned geometries of these compounds. For instance, octahedral complexes are known from SALCs to feature the greatest interaction between the dz2 frontier orbital and the ligands. Relative interaction intensities between the d orbitals and ligands for other geometries, such as square planar and tetrahedral, also are known.
The potential of the d orbitals to interact with the ligands affects the relative energies between them. Due to the toroidal shape of the dz2 orbital, it is capable of interacting with ligands in both the axial and equatorial orientations, resulting in its energy being comparatively raised relative to the other d orbitals due to the repulsive interaction with the ligands. Furthermore, in tetrahedral and octahedral complexes, the stronger overall interaction of the dz2 and dx2-y2 orbitals with the ligands induces the separation of these two d orbitals from the dxz, dyz, and dxy orbtials. This results in d orbital splitting into a doubly degenerate (eg) state and a triply degenerate (t2g) state. The energy splitting of the two d orbital states is a predictor of electron spin state, stability, and luminescent color emitted by the complex in solution. Electrostatic theory can be used to predict electronic transitions between the ground-state t2g and excited state eg energy levels. Complexes in which electrons fill the t2g orbitals prior to filling the eg orbitals are referred to as low-spin complexes; conversely, complexes in which electrons will fill the eg orbitals prior to completely filling the t2g orbitals, thereby maximizing the number of electron parallel spins, are referred to as high-spin complexes.
Molecular orbital (MO) theory uses a linear combination of atomic orbitals (LCAO) to represent molecular orbitals resulting from bonds between atoms. These are often divided into bonding orbitals, anti-bonding orbitals, and non-bonding orbitals. The Schrödinger equation—an application of the wave equation to quantum mechanics used to build up the orbital bases of an atom or molecule—can be solved in three-dimensions to determine the shape of each molecular orbital. A molecular orbital is a Schrödinger orbital that includes two or more nuclei. If this orbital is of the type in which the electron(s) in the orbital have a higher probability of being between nuclei than elsewhere, the orbital will be a bonding orbital, and will tend to hold the nuclei together. A bonding orbital is of lower energy than the two contributing atomic bases, and the percentage composition of each atomic orbital to the molecular orbital is determined by molecular orbital theory. For bonding orbitals, the atomic orbital of lower energy contributes more to the molecular orbital, causing the molecular orbital to more closely resemble it. If the electrons tend to be present in a molecular orbital in which they spend more time elsewhere than between the nuclei, the orbital will function as an anti-bonding orbital and will actually weaken the bond. Conversely, an antibonding orbital will more closely resemble the higher energy atomic basis. Electrons in non-bonding orbitals tend to be in deep orbitals (nearly atomic orbitals) associated almost entirely with one nucleus or the other, and thus they spend equal time within as they do between nuclei. These electrons neither contribute to nor detract from bond strength and thus are best described as lone pairs.
Molecular orbitals are further divided according to the types of atomic orbitals combining to form a bond. These orbitals are results of electron-nucleus interactions that are caused by the fundamental force of electromagnetism—the attraction between the positively charged nucleus and negatively charged electron.
Quantum mechanics stipulates that an electron's position is confined solely to the atomic orbital basis, characterized by s, p, d and f subshells with unique shielding parameters. Chemical substances will form a bond if their orbitals become lower in energy when they interact with each other. This decrease in energy may stem from pi backbonding of a ligand to a metal, which can stabilize a metal's electron pair through delocalization. Thus, bonding molecular orbitals have greater electron density than their component atomic orbitals. Different chemical bonds are distinguished that differ by electron configuration (electron cloud shape) and by energy levels. All atomic orbitals are characterized by a principal atomic number n, or energy level, an azimuthal quantum number l distinguishing s, p, d and f subshells, and a magnetic quantum number m defined as the range –l<m<l. In addition, all electrons are placed into orbitals and are assigned appropriate spin according to the Aufbau principle, Hund's rule and the Pauli exclusion principle.
Molecular orbital diagrams are how chemists analyze the electron configuration of a molecule. They are constructed as qualitative descriptions of bonding within molecules by invoking Molecular Orbital Theory. While VSEPR and valence bond theory are applied to predict the structures of molecules, MO theory is used to determine orbital interaction and valence electron configuration within the molecular orbitals of a molecule. MO diagrams incorporate the wavelike behavior of electrons as postulated by MO theory. They are used to predict the shape of a molecule, as well as the energy, length and angle of each bond. Upon constructing an MO diagram, the following stipulations must be met:
- Orbitals are built up along the y-axis from lowest to highest energy.
- Atomic orbitals of the individual atoms are drawn to the left and right of the molecular orbitals, and are traced to each MO they contribute to.
- Overlapping atomic orbitals contribute to molecular orbitals that possess σ, π, and/or nonbonding character.
- Valence electrons from the atomic orbitals are assigned to molecular orbitals by the Pauli Exclusion Principle.
A complete MO diagram therefore reveals the transformation of all the atomic orbitals into their resulting molecular orbital descriptions. MO diagrams are filled by first determining the number of valence electrons contributed by all the atoms of a molecule, and then evaluating the strength of the orbital interactions by the degree of overlap. Because σ bonds feature greater overlap than π bonds, which are usually long-range, σ and σ* bonding and antibonding orbitals, respectively, feature greater energy splitting (separation) than π and π* orbitals.
MO theory provides a global, delocalized perspective on chemical bonding. In MO theory, any electron in a molecule may be found anywhere in the molecule, since quantum conditions allow electrons to travel under the influence of an arbitrarily large number of nuclei, as long as they are in eigenstates permitted by certain quantum rules. Thus, when excited with the requisite amount of energy through high-frequency light or other means, electrons can transition to higher-energy—even antibonding—molecular orbitals. For instance, in the simple case of a hydrogen diatomic molecule, promotion of a single electron from a bonding orbital to an antibonding orbital can occur under UV radiation. This promotion weakens the bond between the two hydrogen atoms and can lead to photodissociation—the breaking of a chemical bond due to the absorption of light.
Although in MO theory some molecular orbitals may hold electrons that are more localized between specific pairs of molecular atoms, other orbitals may hold electrons that are spread more uniformly over the molecule. Thus, overall, bonding is far more delocalized in MO theory, which makes it more applicable to resonant molecules that have equivalent non-integer bond orders than valence bond (VB) theory. For instance, Missouri theory is used to explain the resonant intermediate bond character of NO3-, in which the N-O bonds have a mix of single and double bond character. This makes MO theory more useful for the description of extended systems.
An example is the MO description of benzene, C
6, which is an aromatic hexagonal ring of six carbon atoms and three double bonds. In this molecule, 24 of the 30 total valence bonding electrons—24 coming from carbon atoms and 6 coming from hydrogen atoms—are located in 12 σ (sigma) bonding orbitals, which are located mostly between pairs of atoms (C-C or C-H), similarly to the electrons in the valence bond description. However, in benzene the remaining six bonding electrons are located in three π (pi) molecular bonding orbitals that are delocalized around the ring. Two of these electrons are in an MO that has equal resonant contributions from all six atoms. The other four electrons are in orbitals with vertical nodes at right angles to each other. As in the VB theory, all of these six delocalized π electrons reside in a larger space that exists above and below the ring plane. All carbon-carbon bonds in benzene are chemically equivalent. In MO theory this is a direct consequence of the fact that the three molecular π orbitals combine and evenly spread the extra six electrons over six carbon atoms.
In molecules such as methane, CH
4, the eight valence electrons are found in four MOs that are spread out over all five atoms. However, it is possible to approximate the MOs with four localized orbitals similar in shape to the sp3 hybrid orbitals predicted by VB theory. Linus Pauling, in 1931, hybridized the carbon 2s and 2p orbitals so that they pointed directly at the hydrogen 1s basis functions and featured maximal overlap. This is often adequate for σ bonds, but is not possible for the π orbitals. However, the delocalized MO description is more appropriate for ionization and spectroscopic predictions. When methane is ionized, a single electron is taken from the MO, which surrounds the whole molecule, weakening all four bonds equally. VB theory would predict that one electron is removed from an sp3 orbital, resulting in the need for resonance between four valence bond structures, each of which has a single one-electron bond and three two-electron bonds. Thus, while the VB model predicts only one ionization energy for methane, there are two ionization energies from the lowest molecular orbitals as predicted by MO theory.
As in benzene, in substances such as beta carotene, chlorophyll, or heme, some electrons in the π orbitals are spread out in molecular orbitals over long distances in a molecule, resulting in light absorption in lower energies (the visible spectrum), which accounts for the characteristic colours of these substances. This and other spectroscopic data for molecules are better explained in MO theory, with an emphasis on electronic states associated with multicenter orbitals, including mixing of orbitals premised on principles of orbital symmetry matching. The same MO principles also more naturally explain some electrical phenomena, such as high electrical conductivity in the planar direction of the hexagonal atomic sheets that exist in graphite. This results from continuous band overlap of half-filled p orbitals. In MO theory, "resonance" is a natural consequence of symmetry. For example, in graphite, as in benzene, it is not necessary to invoke the sp2 hybridization and resonance of VB theory, in order to explain electrical conduction. Instead, MO theory recognizes that some electrons in the graphite atomic sheets are completely delocalized over arbitrary distances, and reside in very large molecular orbitals that cover an entire graphite sheet, and some electrons are thus as free to move and therefore conduct electricity in the sheet plane, as if they resided in a metal.
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